Vadim Ponomarenko

Multivariate Analysis of Functional Metagenomes

Metagenomics is a primary tool for the description of microbial and viral communities. The sheer magnitude of the data generated in each metagenome makes identifying key differences in the function and taxonomy between communities difficult to elucidate. Here we discuss the application of seven different data mining and statistical analyses by comparing and contrasting the metabolic functions of 212 microbial metagenomes within and between 10 environments. Not all approaches are appropriate for all questions, and researchers should decide which approach addresses their questions. This work demonstrated the use of each approach: for example, random forests provided a robust and enlightening description of both the clustering of metagenomes and the metabolic processes that were important in separating microbial communities from different environments. All analyses identified that the presence of phage genes within the microbial community was a predictor of whether the microbial community was host-associated or free-living. Several analyses identified the subtle differences that occur with environments, such as those seen in different regions of the marine environment.

The Rényi-Ulam pathological liar game with a fixed number of lies

The q-round Renyi-Ulam pathological liar game with k lies on the set [n] := {1,....,n} is a 2-player perfect information zero sum game. In each round Paul chooses a subset A ⊆ [n] and Carole either assigns 1 lie to each element of A or to each element of [n]\A. Paul wins if after q rounds there is at least one element with k or fewer lies. The game is dual to the original Renyi-Ulam liar game for which the winning condition is that at most one element has k or fewer lies. Define Fk*(q) to be the minimum n such that Paul can win the q-round pathological liar game with k lies and initial set [n]. For fixed k we prove that Fk*(q) is within an absolute constant (depending only on k) of the sphere bound, 2q/ (q ≥k); this is already known to hold for the original Renyi-Ulam liar game due to a result of J. Spencer.

Faà di Bruno's formula and nonhyperbolic fixed points of one-dimensional maps

Fixed-point theory of one-dimensional maps of ℝ does not completely address the issue of nonhyperbolic fixed points. This note generalizes the existing tests to completely classify all such fixed points. To do this, a family of operators are exhibited that are analogous to generalizations of the Schwarzian derivative. In addition, a family of functions f are exhibited such that the Maclaurin series of f ( f ( x ) ) and x are identical.

Numerical Semigroups on Compound Sequences

We generalize the geometric sequence {ap, ap−1b, ap−2b2,…, bp} to allow the p copies of a (resp. b) to all be different. We call the sequence {a1a2a3… ap, b1a2a3…ap, b1b2a3…ap,…, b1b2b3…bp} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.

Two Semigroup Elements Can Commute With Any Positive Rational Probability

Summary We construct semigroups with any given positive rational commuting probability, extending a Classroom Capsule from November 2008 in this Journal.

Irreducible Factorization Lengths and the Elasticity Problem within ℕ

Abstract A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism). For monoids in this class, the important factorization property of finite elasticity is characterized.

Robert Gilmer’s contributions to the theory of integer-valued polynomials

denote the ring of integer-valued polynomials on D with respect to the subset E (for ease of notation, if E = D, then set Int(D,D) = Int(D)). Gilmer’s work in this area (with the assistance of various co-authors) was truly groundbreaking and led to numerous extensions and generalizations by authors such as J. L. Chabert, P. J. Cahen, D. McQuillan and A. Loper. In this paper, we will review Gilmer’s papers dedicated to this subject. We close with an elementary analysis of polynomial closure in integral domains, a topic which Gilmer motivated with a characterization of which subsets S of Z define the ring Int(Z) in [18]. Before proceeding, please note that we use Q to represent the rationals, Z the integers, N the natural numbers and P the primes in Z. It is clear that Gilmer’s interest in the rings Int(E,D) was motivated by his early work on multiplicative ideal theory and the theory of Prufer domains. In particular, there was a problem open in the early 60’s regarding the number of required generators for a finitely generated ideal of a Prufer domain. It was at the time well known in the Noetherian case (i.e., for a Dedekind domain) that every ideal could be generated by two elements, one of which could be chosen to be an arbitrary non-zero element of the ideal.

Subsums of the Harmonic Series

Abstract We consider subsums of the harmonic series, and determine conditions for their convergence. We apply these conditions to determine convergence for a family of series that generalizes Kempner’s series.

Arithmetic of Congruence Monoids

Let ℕ represent the positive integers. Let n ∈ ℕ and Γ ⊆ ℕ. Set Γn = {x ∈ ℕ: ∃ y ∈ Γ, x ≡ ymodn} ∪ {1}. If Γn is closed under multiplication, it is known as a congruence monoid or CM. A classical result of James and Niven [15] is that for each n, exactly one CM admits unique factorization into products of irreducibles, namely Γn = {x ∈ ℕ: gcd (x, n) = 1}. In this article, we examine additional factorization properties of CMs. We characterize CMs that contain primes, and we determine elasticity for several classes of CMs and bound it for several others. Also, for several classes, we characterize half-factoriality and determine whether the elasticity is accepted and whether it is full.